Question

1-2sin^2x+cos(4x-2п)

Answer 1

1-2sin^2x+cos(4x-2п)=cos2x+cos4x=2cos(3x)cosx

Answer 2

To simplify the expression $1-2\sin^2x + \cos(4x-2\pi)$, we can use trigonometric identities to rewrite and simplify it.

1. Recall the double angle identity for cosine: $\cos(2\theta) = 1 - 2\sin^2\theta$.

2. We know that $\cos(\theta + \pi) = -\cos(\theta)$. Therefore, $\cos(4x-2\pi) = -\cos(4x)$.

Now, let's substitute these identities into the original expression:

$1 - 2\sin^2x + \cos(4x-2\pi)$

$= 1 - 2\sin^2x - \cos(4x)$

Now, we can use the double angle identity for cosine to rewrite $\cos(4x)$:

$= 1 - 2\sin^2x - (1 - 2\sin^2(2x))$

$= 1 - 2\sin^2x - 1 + 4\sin^2x$

Combine like terms:

$= 4\sin^2x - 2\sin^2x$

$= 2\sin^2x$

So, the simplified expression is $2\sin^2x$.